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Some Integral Inequalities of the Hermite-Hadamard Type for Strongly Quasi-convex Functions

Yi-Xuan Sun, Jing-Yu Wang, Bai-Ni Guo*

School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, China

Abstract: In the paper, the authors introduce a new notion “strongly quasi-convex function”, establish an integral identity for strongly quasi-convex functions, and establish some new integral inequalities of the Hermite-Hadamard type for strongly quasi-convex functions.

Keywords: integral identity, integral inequality, Hermite-Hadamard type, strongly quasi-convex function, Hölder inequality

Citation: Yi-Xuan Sun, Jing-Yu Wang, Bai-Ni Guo, Some Integral Inequalities of the Hermite-Hadamard Type for Strongly Quasi-convex Functions. Turkish Journal of Analysis and Number Theory, 4 (5). 132-134. January 2016. doi:10.12691/tjant-4-5-2

1. Introduction

We first list some definitions concerning various convex functions.

Definition 1.1. A function is said to be convex if

[picture] | (1.1)

holds for alland.

Deﬁnition 1.2 ([1]). A function is said to be quasi-convex convex if

[picture] | (1.2)

for all

Deﬁnition 1.3 ([2]). A function is said to be strongly convex with modulus if

[picture] | (1.3)

is valid for all and .

The following inequalities of the Hermite-Hadamard type were established for the above convex functions.

Theorem 1.1 ([3]). Let be differentiable on and with .

(i) If is convex function on , then

[picture] | (1.4)

(ii) If is convex function on and , then

[picture] | (1.5)

In [4], two inequalities of the Hermite-Hadamard type for quasi-convex functions were introduced as follow.

Theorem 1.2 ([4]). Let be a differentiable mapping on and with . If is quasi-convex on , then

[picture] | (1.6)

Theorem 1.3 ([4]). Let be a differentiable mapping on and with . If is quasi-convex on and , then

[picture] | (1.7)

Theorem 1.4 ([5, Theorem 2.3]). Let be a differentiable mapping on and with . If is quasi-convex on and , then

[picture] | (1.8)

Theorem 1.5 ([5, Theorem2.4]). Let be a differentiable mapping on and with . If is quasi-convex on and , then

[picture] | (1.9)

In this paper, we will introduce a new notion “strongly quasi-convex function” and establish some integral inequalities of the Hermite-Hadamard type for functions whose derivatives are of strongly quasi-convexity.

2. A Definition and a Lemma

We now introduce the notion “strongly quasi-convex functions”.

Deﬁnition 2.1. A function is said to be strongly quasi-convex with modulus if

is valid for all and .

For establishing new integral inequalities of the Hermite--Hadamard type for strongly quasi-convex functions, we need the following identity.

Lemma 2.1. Let be differentiable on and with . If , then

[picture] | (2.1)

Proof. This follows from a straightforward computation of definite integrals.

3. Some Integral Inequalities of the Hermite-Hadamard Type

Now we are in a position to establish some integral inequalities of the Hermite–-Hadamard type for functions whose derivatives are of strongly quasi-convexity.

Theorem 3.1. Let be differentiable mapping on and with . If and is strongly quasi-convex on for with modulus , then

Proof. Since is strongly quasi-convex on , using Lemma 2.1 and by Hölder’s inequality, we have

Theorem 3.1 is thus proved.

Crollary 3.1. Under conditions of Theorem 3.1, if , then

Theorem 3.2. Let be differentiable mapping on and with . If and is strongly quasi-convex on for with modulus , then

Proof. By Lemma 2.1 and using Hölder’s inequality and the strongly quasi-convexity of , we obtain

Theorem 3.3 is proved.

Acknowledgement

This work was partially supported by the National Natural Science Foundation under Grant No. 11361038 of and by the Inner Mongolia Autonomous Region Natural Science Foundation Project under Grant No. 2015MS0123, .

The authors thank the anonymous referee for his/her careful corrections to and valuable comments on the original version of this paper.

References

[1]   | Dragomir, S. S., Pečarić, J., and Persson, L. E., Some inequalities of Hadamard type, Soochow J. Math., 21 (3) (1995), 335-341.
[2]   | Polyak, B. T., Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Soviet Math. Dokl., 7 (1966), 72-75.
[3]   | Dragomir, S. S. and Agarwal, R. P., Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula, Appl. Math. Lett., 11 (1998), 91-95.
[4]   | Ion, D. A., Some estimates on the Hermite--Hadamard inequality through quasi-convex functions, Ann. Univ. Craiova Math. Comp. Sci. Ser., 34 (2007), 82-87.
[5]   | Alomari, M., Darus, M., and Kirmaci, U. S., Refinements of Hadamard-type inequalities for quasi-convex functions with applications to trapezoidal formula and special means, Comput. Math. Appl., 59 (2010), 225-232.